Question

The differential equation of all parabolas whose axis is y-axis is:

• A. $x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$
• B. $x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=0$
• C. $\frac{d^{2}y}{d{x}^2}-y=0$
• D. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$

Answer 1

The correct option is A $x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$

$axis=yaxis$

vertex is $(0,k)$
Equation of parabola is
$(x-0{)}^2=4a(y-k)$
$x^2=4ay-4ak$
Differentiate w.r.t $x$
$2x=4a\frac{dy}{dx}$
$x=2a\frac{dy}{dx}$
$\therefore \frac{1}{2a}=\frac{1}{x}\frac{dy}{dx}$

Differentiate w.r.t $x$,
$\frac{d}{dx}(\frac{1}{x}.\frac{dy}{dx})=\frac{d}{dx}\left(\frac{1}{2a}\right)$
$\frac{1}{x}.\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}(-\frac{1}{x^2})=0$
$\therefore x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$